Let \( W \) be the number of women and \( M \) be the number of men on the train at station A. According to the given ratio, we have:
\[ \frac{W}{M} = \frac{3}{5} \]
This means we can express \( W \) and \( M \) in terms of a common variable \( k \):
\[ W = 3k \quad \text{and} \quad M = 5k \]
At station B, \( \frac{1}{4} \) of the men alight from the train:
\[ \text{Men alighting} = \frac{1}{4} M = \frac{1}{4} (5k) = \frac{5k}{4} \] \[ \text{Men remaining} = M - \text{Men alighting} = 5k - \frac{5k}{4} = \frac{20k}{4} - \frac{5k}{4} = \frac{15k}{4} \]
Let \( x \) be the number of women who alight from the train at station B. The number of women remaining now is:
\[ W - x = 3k - x \]
The new ratio of women to men becomes \( \frac{8}{15} \):
\[ \frac{3k - x}{\frac{15k}{4}} = \frac{8}{15} \]
Cross-multiplying gives:
\[ 15(3k - x) = 8 \left(\frac{15k}{4}\right) \]
Simplifying the right-hand side:
\[ 15(3k - x) = 30k \]
This expands to:
\[ 45k - 15x = 30k \]
Rearranging the equation yields:
\[ 15k = 15x \quad \Rightarrow \quad x = k \]
Thus, the number of women who alighted at station B is \( k \).
Part (B): Total Passengers after Station C
Now we investigate the situation at station C. The total number of women remaining after station B:
\[ 3k - x = 3k - k = 2k \]
The total number of men remaining after station B:
\[ \frac{15k}{4} \]
At station C, 100 women and 30 men boarded the train. Therefore, the new number of women and men is:
\[ \text{New number of women} = 2k + 100 \] \[ \text{New number of men} = \frac{15k}{4} + 30 \]
After station C, the new ratio of women to men becomes equal:
\[ 2k + 100 = \frac{15k}{4} + 30 \]
Multiplying the entire equation by 4 to eliminate the fraction gives:
\[ 4(2k + 100) = 15k + 120 \]
Expanding this results in:
\[ 8k + 400 = 15k + 120 \]
Rearranging the equation leads to:
\[ 400 - 120 = 15k - 8k \quad \Rightarrow \quad 280 = 7k \quad \Rightarrow \quad k = 40 \]
We now know \( k \) and can find the numbers of women and men after station C:
- Number of women after B:
\[ 2k = 2 \times 40 = 80 \]
- Number of men after B:
\[ \frac{15k}{4} = \frac{15 \times 40}{4} = 150 \]
Adding the new passengers boarding at station C:
- Total women after C:
\[ 80 + 100 = 180 \]
- Total men after C:
\[ 150 + 30 = 180 \]
Total passengers after station C:
\[ \text{Total passengers} = 180 + 180 = 360 \]
Final Answers
(A) The total number of passengers on the train after Station C is \( \mathbf{360} \).
(B) The number of women who alighted at Station B is \( \mathbf{40} \).